English

A random variable X has the following probability distribution : x = x 0 1 2 3 7 P(X=x) 0 k 2k 2k 3k k2 2k2 7k2 + k Determine (i) k (ii) P(X> 6) (iii) P(0<X<3). - Mathematics and Statistics

Advertisements
Advertisements

Question

A random variable X has the following probability distribution :

x = x 0 1 2 3       7
P(X=x) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine (i) k

(ii) P(X> 6)

(iii) P(0<X<3).

Sum
Advertisements

Solution

Refer to the solution of Q. 8 of Exercise 7.1.

shaalaa.com
  Is there an error in this question or solution?
Chapter 7: Probability Distributions - Miscellaneous Exercise 2 [Page 244]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
Chapter 7 Probability Distributions
Miscellaneous Exercise 2 | Q 8 | Page 244

RELATED QUESTIONS

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.


State the following are not the probability distributions of a random variable. Give reasons for your answer.

Y -1 0 1
P(Y) 0.6 0.1 0.2

Find the probability distribution of number of heads in two tosses of a coin.


Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as

(i) number greater than 4

(ii) six appears on at least one die


A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.


The random variable X has probability distribution P(X) of the following form, where k is some number:

`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`

  1. Determine the value of 'k'.
  2. Find P(X < 2), P(X ≥ 2), P(X ≤ 2).

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find E(X).


Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.


Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution.


Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)

X : 3 2 1 0 −1
(X) : 0.3 0.2 0.4 0.1 0.05
 
(ii)
X : 0 1 2
P (X) : 0.6 0.4 0.2


(iii)

X : 0 1 2 3 4
P (X) : 0.1 0.5 0.2 0.1 0.1
 


(iv)

X : 0 1 2 3
P (X) : 0.3 0.2 0.4 0.1
 

A random variable X has the following probability distribution:

Values of X : −2 −1 0 1 2 3
P (X) : 0.1 k 0.2 2k 0.3 k
 

Find the value of k


Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P (X = x1) = 3P(X = x2) = P (X = x3) = 5 P (X = x4). Find the probability distribution of X.                                                                                                                                                                                 


A bag contains 4 red and 6 black balls. Three balls are drawn at random. Find the probability distribution of the number of red balls.


A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?


Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls


From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement .

 

Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.


A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.


Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?


The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

Determine the value of k .


The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

 Find P(X ≤ 2) + P(X > 2) .

 

Find the mean variance and standard deviation of the following probability distribution 

xi : a b
pi : p q
where p + q = 1 .

A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.


Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution .  


Find the mean of the following probability distribution:

Xxi: 1 2 3
P(Xxi) :
\[\frac{1}{4}\]
 
\[\frac{1}{8}\]
\[\frac{5}{8}\]

 


If the probability distribution of a random variable X is as given below:

Write the value of P (X ≤ 2).

X = xi : 1 2 3 4
P (X = xi) : c 2c 4c 4c

 

 

If a random variable X has the following probability distribution:

X : 0 1 2 3 4 5 6 7 8
P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a

then the value of a is


An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.


Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.


Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.


Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).


The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )


Write the negation of the following statements : 

(a) Chetan has black hair and blue eyes. 
(b) ∃ x ∈ R such that x2 + 3 > 0. 


Determine whether each of the following is a probability distribution. Give reasons for your answer.

x 0 1 2
P(x) 0.4 0.4 0.2

Determine whether each of the following is a probability distribution. Give reasons for your answer.

x 0 1 2 3 4
P(x) 0.1 0.5 0.2 –0.1 0.3

Solve the following problem :

If a fair coin is tossed 4 times, find the probability that it shows 3 heads


Solve the following problem :

A large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Find the probability that the inspector finds at most one defective item in the 4 selected items.


Solve the following problem :

It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of a week.


Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X


The probability distribution of a random variable X is given below:

X 0 1 2 3
P(X) k `"k"/2` `"k"/4` `"k"/8`

Determine P(X ≤ 2) and P(X > 2)


Two probability distributions of the discrete random variable X and Y are given below.

X 0 1 2 3
P(X) `1/5` `2/5` `1/5` `1/5`

 

Y 0 1 2 3
P(Y) `1/5` `3/10` `2/10` `1/10`

Prove that E(Y2) = 2E(X).


The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P(X = 1) = p and that E(X2) = E[X], find the value of p


Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate Standard deviation of X.


The probability distribution of a random variable x is given as under:
P(X = x) = `{{:("k"x^2,  "for"  x = 1"," 2"," 3),(2"k"x,  "for"  x = 4"," 5"," 6),(0,  "otherwise"):}`
where k is a constant. Calculate E(3X2)


A random variable x has to following probability distribution.

X 0 1 2 3 4 5 6 7
P(x) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine


Find the mean of number randomly selected from 1 to 15.


A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×